Microscopy
Femius Koenderink Center for Nanophotonics FOM Institute AMOLF Amsterdam
1. Microscopy at the diffraction limit - What is the diffraction limit - Spatial frequencies and Fourier transforms - High NA imaging
2. Microscopy beyond the diffraction limit - Scanning probe to beat the diffraction limit - Example imaging molecules - Example imaing photonic structure - Artefacts
Arbitrary source distribution
( ) ( ) i2
1, ; , , e d d4
x yk x k yx yk k z x y z x y
π − +
−∞
= ∫∫E E
Describe field as superposition of plane waves (Fourier transform):
( ) ( ) iˆ, , , ; e d dx yk x k yx y x yx y z k k z k k +
−∞
=
∫∫E E
E∞
∞
This representation is called ‘Angular spectrum representation’
Arbitrary source distribution
( ) ( ) i2
1, ; , , e d d4
x yk x k yx yk k z x y z x y
π − +
−∞
= ∫∫E E
Describe field as superposition of plane waves (Fourier transform):
( ) ( ) iˆ, , , ; e d dx yk x k yx y x yx y z k k z k k +
−∞
=
∫∫E E
Field at z=0 (object) propagates in free space as
( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2
0z x yk nk k k= − +
E∞
∞
Arbitrary source distribution
Field at z=0 (object) propagates in free space as
( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2
0z x yk nk k k= − +
The propagator is oscillating for
and exponentially decaying for
( ) ( )22 20x yk k nk+ <
( ) ( )22 20x yk k nk+ >
Near field = Exponentially confined fast spatial fluctuations Far field = Propagating fields = bound by diffraction limit
Now find the field along the beam Insert the Gaussian into
Paraxial approximation If only small angles contribute
Again we are transforming a Gaussian
Gaussian beam optics
A gaussian object results in a gaussian beam as far field Diffraction: the beam widens away from the waist
Gaussian beam optics
Diffraction: the beam widens away from the waist The narrower the waist, the more the divergence Note how the law Relates spot size and numerical aperture NA=n sinθ
Diffraction optics intuition
1) Narrow beams lead to larger angular divergence 2) Larger beams can hence be more tightly focused
3) Angular far field profile to first order is just the Fourier transform of the source distribution - Gaussian beam - Also: diffraction by pinholes, and slits.
Arbitrary source distribution
Field at z=0 (object) propagates in free space as
( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2
0z x yk nk k k= − +
The propagator H is oscillating for
and exponentially decaying for
( ) ( )22 20x yk k nk+ <
( ) ( )22 20x yk k nk+ >
Near field = Exponentially confined fast spatial fluctuations Far field = Propagating fields = bound by diffraction limit Small objects <> wide beams and high NA’s
The diffraction limit Image of a point source in a microscope, collecting part of the angular spectrum of the source:
Rayleigh criterion: two point sources distinguishable if spaced by the distance between the maximum and the first minimum of the Airy pattern
+ θ sinNA n θ=
The diffraction limit Image of a point source in a microscope, collecting part of the angular spectrum of the source:
Rayleigh criterion: two point sources distinguishable if spaced by the distance between the maximum and the first minimum of the Airy pattern
+
0.61dNAλ
=
θ
sinNA n θ=
Rigorous non paraxial calculation gives 0.61 from Airy pattern
What’s in a rigorous calculation
Compare normal ray optics: lenses approximated as planes Abbe sine condition is `holy design rule’ for microscopes
Abbe sine condition High NA: hemispherical reference surface
What’s in a rigorous calculation
Abbe sine condition `aplanatic lens’ High NA: 1. hemispherical reference surface 2. constant power in rays upon crossing reference 3. Upon refraction, polarization vector refracts too Strategy works for illumination and collection geometry
Important consequences
Polarized as incident
Polarized along the beam
Best focus is λ/2NA in size Strong focusing adds polarization out-of-plane Focus is not quite cylindrical in shape due to polarization
High NA imaging
The ultimate smallest object is a molecule
Z-dipole In-plane Tilted 45 deg
NA=1.4 In focus
NA=0.4 In focus
Fourier microscopy
Back aperture directly maps sinθ ~k||
objective (NA=0.95)
back aperture Supercontinuum light source (Fianium) + AOTF
λ=600 nm 5 µm
Fourier microscopy
Direct evidence of k|| + G conservation
objective (NA=0.95)
back aperture Supercontinuum light source (Fianium) + AOTF
λ=600 nm
Single scatterer radiation patterns
21
Cts/pxl (0.1 s)
In phase-excited plasmon rods radiate like a line of dipoles: donut x sinc function Potential: visualize radiation pattern of any SINGLE antenna
0
1200
ky
kx
(λ=600 nm)
1um x 100 nm Au
0
500
ky
kx
(λ=600 nm)
2um x 100 nm Au
Confocal microscope
Highest resolution imaging with lenses 1. Overfill high NA objective with a parallel beam 2. Color separate output at dichroic element 3. Tube lens to focus on point detector
Resolution from confocality: (1) small laser spot (2) detection pinhole
Confocal microscope
Typical numbers 100x, NA =0.95 – 1.4 Tube lens L1/L2: f2=200 mm Objective f1 = 200/100 = 2mm Beam radius f1 NA = 3 mm Note also `conservation rule’
Resolution object plane
Gaussian optics
Why NA really helps
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
Objective opening angle (degrees)90756030 450
NA=0.4 4%
NA=0.95 35%
NA=0.9 28%
Capt
ured
frac
tion
of 4
π sr
Objective NA
NA=0.7 15%
15
NA means 1. Resolution
2. Detection efficiency
Seeing single molecules
Very dilute layer of fluorescing Molecules (DiD)
Single molecules blink one by one In: 532 nm, ~ 0.1 mW Out: ~ 600 nm
Courtesy: Martin Frimmer, AMOLF open dag
Enabling equipment tricks For room temperature experiments the universal tricks are: Dilution - λ/2 detection or excitation volume - Diluted samples to < 1 molecule per 1 µm2
-Filtering - 108 laser line rejection filters Efficient photon collection - Very high NA objective
Shot noise level detection - Silicon CCDs and APDs 60% q.efficiency
low read out & dark noise
1. Microscopy at the diffraction limit - What is the diffraction limit - Spatial frequencies and Fourier transforms - High NA imaging
2. Microscopy beyond the diffraction limit - Scanning probe to beat the diffraction limit - Example imaging molecules - Example imaing photonic structure - Artefacts
Localization of a molecule Idea: if you know you have a single object, you can find its
localization to much better than the diffraction limit
Single molecule CCD images Different exposure times
Least square fitting of Gaussian Error diminishes with photon count N
Beam waist Pixel size Background noise
Biophysical Journal 82(5) 2775–2783
Note: not true diffraction barrier breaking
Cheating the diffraction limit
PALM, STORM: beat Abbe limit by seeing a single molecule at a time Using a stochastic on/off switch to keep most molecules dark
Resolution: how discernible are two objects ? If you have a single object, you can fit the center of a Gaussian with arbitrary precision (depends on noise)
Arbitrary source distribution
Field at z=0 (object) propagates in free space as
( ) ( ) iˆ ˆ, ; , ;0 e zk zx y x yk k z k k ±=E E ( ) ( )2 2 2
0z x yk nk k k= − +
The propagator H is oscillating for
and exponentially decaying for
( ) ( )22 20x yk k nk+ <
( ) ( )22 20x yk k nk+ >
Near field = Exponentially confined fast spatial fluctuations Far field = Propagating fields = bound by diffraction limit
Field of a dipole
1/r term: far field constant radiated flux
1/r3 term: near field 1/r2 term: mid field
Transition near to far at kr ~1 or r ~λ/2π
Breaking the diffraction limit in near-field microscopy
A small aperture in the near field of the source can scatter also the evanescent field of the source to a detector in the far field.
Image obtained by scanning the aperture
Alternatively, the aperture can be used to illuminate only a very small spot.
Aperture probe fibre type
Aperture probe microlever type
Metallic particle Single emitter
Probing beyond the diffraction limit
glass
aluminum
500 nm
100 nm
100 nm
λ
35 nm aperture
– well defined aperture – flat endface – isotropic polarisation – high brightness up 1 µW
Ex Ey Ez
With excitation Ex , kz, :
Focussed ion beam (FIB) etched NSOM probe
Veerman, Otter, Kuipers, van Hulst, Appl. Phys. Lett. 74, 3115 (1998)
x
y
Shear force feedback: molecular scale topography
Feedback on phase Tip -sample < 5 nm RMS ~ 0.1 nm
Feedback loop:
sample
Lateral movement, A0 ~ 0.1 nm
Tuning fork 32 kHz Q ~ 500
∆f
ω0
A0
piezo
Rensen, Ruiter, West, van Hulst, Appl. Phys. Lett. 75 1640 (1999) Ruiter, Veerman, v/d Werf, van Hulst, Appl. Phys. Lett. 71 28 (1997)
van Hulst, Garcia-Parajo, Moers, Veerman, Ruiter, J. Struct. Biol. 119, 222, (1997)
1.7 x 1.7 µm
3 x 3 µm
Steps on graphite (HOPG)
~ 0.8 nm step ~ 3 mono-atomic steps
DNA width 14 nm
height 1.4 nm
DNA on mica
90o 0o 1 µm
100 nm
Perylene orange in PMMA
Ruiter, Veerman, Garcia-Parajo, van Hulst, J. Phys. Chem. 101 A, 7318
Two arms of the interferometer
equal in length gives temporal overlap on the
detector
Time-resolved near-field scanning tunneling microscopy
Artefacts
1. Topographic The tip moves over the topography Potential cross talk
1. Non-perturbative tip
Topographic artefacts
Topography: convolution of sample and tip Optical: weighted by exponential factor Tricky: topography and optical pick up are shifted sideways
43
Narrow cavity resonance
Laser: grating tunable diode laser 20 MHz linewidth around 1565 nm Detection: InGaAs APD (IdQuantique)
1565.0 1565.2 1565.40
25
50
75
100
125
Coun
ts
Wavelength (nm)
Picked up by tipFew µm above cavity Q =(1±0.5) 105
Lorentz Q =88000
November
44
Resonance shift
1565.0 1565.2 1565.40
25
50
75
100
125
Coun
ts
Wavelength (nm)
Few µm above cavity~10 nm above cavity
Line shifts by 1 linewidth
Glass tip: ∆ω/ω ∼ −1.2 10−4 (∆λ of 20 pm)
Consistent with
2 /0
2mode 0
| ( ) |max( ( ) | ( ) | )
z dE r eV E
ω αω ε
−∆= − ⋅
r r
Inserting a polarizability comparable to the mode volume shifts ω
45
Tuning vs mode intensity
1565.20
1565.25
1565.30
1565.35
1565.40
1565.45
Transverse to W1Along W1
λ of
max
. sig
nal (
nm)
-2 -1 0 1 2-4 -2 0 2 40.0
0.5
1.0
g
|E|2 (n
orm
.) in
the
slab
Position (µm)
Experiment
FDTD
In this case the ∆ω/ω and not Intensity maps |E|2